The key challenge in the above argument is to show that an equation in the form of eq:keyclaimWg0 could in fact hold. At first glance, that an inequality such as eq:keyclaimWg0 should hold is rather non-obvious — this is because while the variable \((Z_{\tau } Z_{\tau } - I)(\nabla f(x_{\tau }) - \nabla f(x_{\tau }^{\prime })) \mid {\mathcal {F}}_{\tau -1}\)  is mean-zero, it is subExponential rather than subGaussian. In fact, even in the subGaussian case, given a sequence of random vectors \({\mathbf {x}}_{0},\dots , {\mathbf {x}}_{t-1}\) , such that each \(\mathbb {E}[{\mathbf {x}}_{\tau } \mid {\mathcal {F}}_{\tau - 1}] = 0\) , and that each \({\mathbf {x}}_{\tau } \mid {\mathcal {F}}_{\tau - 1}\)  is norm-subGaussian with parameter \(\sigma _{\tau } \in {\mathcal {F}}_{\tau -1}\)  (which is an appropriate generalization of subGaussianity for vectors, proposed in [1]}), proving a concentration inequality of the form
\(*{\sum _{\tau = 0}^{t-1} {\mathbf {x}}_{\tau }} \approx \tilde{O} \left(\sqrt{\sum _{\tau = 0}^{t-1} \sigma _{\tau }^2}\right)\) 
